History Analytic Elements in Number Theory Story of Primes Perfect Numbers References A Joyful Walk Through Analytic Number Theory: from Classical to Modern Kwan Chung-Hang, Kevin, [email protected] The Chinese University of Hong Kong EPTMT Guest Lecture 7 August, 2017 History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Story Time! Many good stories start with `once upon a time'. Number Theory is one of these stories. More precisely, it started from the ancient Greek. It was influenced heavily by the philosophy of that era. Interestingly or bizaarely, they thought of the meanings of numbers a bit too hard. (More like numerology) History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Perfect Numbers and Pythagoras Pythagoreans equated the `perfect number' 6 to marriage, health, and beauty of integrity and agreement. `Six is a perfect number in itself, and not because God created all things in six days; rather the converse is true. God created all things in six days because the number six is perfect.' History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Perfect Numbers and the Bible (First Origin) St. Augustine wrote about perfect numbers that appearing in the Bible as follows: History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Perfect Numbers and the Bible (First Origin) St. Augustine wrote about perfect numbers that appearing in the Bible as follows: `Six is a perfect number in itself, and not because God created all things in six days; rather the converse is true. God created all things in six days because the number six is perfect.' `Eight is not a perfect number (in fact deficient) and the number eight stands for the eight souls in Noah's Ark: Noah, his three sons and their four wives. The entire human race was originated from these eight souls.' History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Perfect Numbers and the Bible (Second Origin) Alcuin of York claimed that the Second Origin is not perfect because... History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Perfect Numbers and the Bible (Second Origin) Alcuin of York claimed that the Second Origin is not perfect because... `Eight is not a perfect number (in fact deficient) and the number eight stands for the eight souls in Noah's Ark: Noah, his three sons and their four wives. The entire human race was originated from these eight souls.' Please don't ask me further about the philosophical or biblical aspects of perfect numbers. I know nothing about these! History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Courtesy to Wikipedia and John Voight's wonderful article `Perfect Numbers: An Elementary Introduction' on the historical account of perfect numbers. History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Courtesy to Wikipedia and John Voight's wonderful article `Perfect Numbers: An Elementary Introduction' on the historical account of perfect numbers. Please don't ask me further about the philosophical or biblical aspects of perfect numbers. I know nothing about these! P In other words, σ(n) = 2n, where σ(n) := d>0 d. djn Non-example: 8 > 1 + 2 + 4. In fact if σ(n) > 2n, then n is abundant; if σ(n) < 2n, then n is deficient. Why writing in σ? Because σ is a multiplicative function, i.e., if m; n are relatively prime natural numbers, then σ(mn) = σ(m)σ(n). History Analytic Elements in Number Theory Story of Primes Perfect Numbers References OK, Be Serious Now! Background... Definition (Perfect Number) A perfect number is a natural number that is equal to sum of all of its proper divisors. Example: 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14 Non-example: 8 > 1 + 2 + 4. In fact if σ(n) > 2n, then n is abundant; if σ(n) < 2n, then n is deficient. Why writing in σ? Because σ is a multiplicative function, i.e., if m; n are relatively prime natural numbers, then σ(mn) = σ(m)σ(n). History Analytic Elements in Number Theory Story of Primes Perfect Numbers References OK, Be Serious Now! Background... Definition (Perfect Number) A perfect number is a natural number that is equal to sum of all of its proper divisors. Example: 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14 P In other words, σ(n) = 2n, where σ(n) := d>0 d. djn Why writing in σ? Because σ is a multiplicative function, i.e., if m; n are relatively prime natural numbers, then σ(mn) = σ(m)σ(n). History Analytic Elements in Number Theory Story of Primes Perfect Numbers References OK, Be Serious Now! Background... Definition (Perfect Number) A perfect number is a natural number that is equal to sum of all of its proper divisors. Example: 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14 P In other words, σ(n) = 2n, where σ(n) := d>0 d. djn Non-example: 8 > 1 + 2 + 4. In fact if σ(n) > 2n, then n is abundant; if σ(n) < 2n, then n is deficient. History Analytic Elements in Number Theory Story of Primes Perfect Numbers References OK, Be Serious Now! Background... Definition (Perfect Number) A perfect number is a natural number that is equal to sum of all of its proper divisors. Example: 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14 P In other words, σ(n) = 2n, where σ(n) := d>0 d. djn Non-example: 8 > 1 + 2 + 4. In fact if σ(n) > 2n, then n is abundant; if σ(n) < 2n, then n is deficient. Why writing in σ? Because σ is a multiplicative function, i.e., if m; n are relatively prime natural numbers, then σ(mn) = σ(m)σ(n). History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Background|What were known? Theorem (Euler/Euclid) A natural number is an even perfect number if and only if it is of the form 2p−1(2p − 1), where p is a prime such that 2p − 1 is also a prime. Computational evidence: An odd perfect number must be greater than 101500, has at least 101 prime factors and at least 10 distinct prime factors. The largest prime factor is greater than 108. (c.f. Math. Comp. Journal for more!). Does odd perfect number exist? And what are you waiting for ?! History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Conjectures: Your Job Opportunity! Are there infinitely many perfect numbers? (6, 28, 496, 8128, ... ???) And what are you waiting for ?! History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Conjectures: Your Job Opportunity! Are there infinitely many perfect numbers? (6, 28, 496, 8128, ... ???) Does odd perfect number exist? History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Conjectures: Your Job Opportunity! Are there infinitely many perfect numbers? (6, 28, 496, 8128, ... ???) Does odd perfect number exist? And what are you waiting for ?! Sadly not much... Pretty much you know them all from primary school (but highly influential!): 1. Sieve of Eratosthenes: How to find primes in a range. (i.e., Trial Divisions) 2. Fundamental Theorem of Arithmetic: Every natural number great than 1 can be decomposed as a product of primes in a unique way, e.g. 15 = 3 × 5, 120 = 23 × 3 × 5. 3. Euclid's Theorem There are infinitely many primes: 2,3,5,7,11,13,17,19,... History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Another Primitive Notion for Ancient Greek Definition (Prime) A natural number greater than 1 is a prime if its positive divisors are only one and itself. What were known in the ancient Greek? 1. Sieve of Eratosthenes: How to find primes in a range. (i.e., Trial Divisions) 2. Fundamental Theorem of Arithmetic: Every natural number great than 1 can be decomposed as a product of primes in a unique way, e.g. 15 = 3 × 5, 120 = 23 × 3 × 5. 3. Euclid's Theorem There are infinitely many primes: 2,3,5,7,11,13,17,19,... History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Another Primitive Notion for Ancient Greek Definition (Prime) A natural number greater than 1 is a prime if its positive divisors are only one and itself. What were known in the ancient Greek? Sadly not much... Pretty much you know them all from primary school (but highly influential!): 2. Fundamental Theorem of Arithmetic: Every natural number great than 1 can be decomposed as a product of primes in a unique way, e.g. 15 = 3 × 5, 120 = 23 × 3 × 5. 3. Euclid's Theorem There are infinitely many primes: 2,3,5,7,11,13,17,19,... History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Another Primitive Notion for Ancient Greek Definition (Prime) A natural number greater than 1 is a prime if its positive divisors are only one and itself. What were known in the ancient Greek? Sadly not much... Pretty much you know them all from primary school (but highly influential!): 1. Sieve of Eratosthenes: How to find primes in a range. (i.e., Trial Divisions) 3. Euclid's Theorem There are infinitely many primes: 2,3,5,7,11,13,17,19,... History Analytic Elements in Number Theory Story of Primes Perfect Numbers References Another Primitive Notion for Ancient Greek Definition (Prime) A natural number greater than 1 is a prime if its positive divisors are only one and itself.